I went to Disney with a pinched nerve in my neck one year. Depending on how careful you need to be, you should be able to manage
most rides.
Space Mountain is horrible for people with bad necks. Stay far, far away!
Primeval Whirl has too many sudden jerks to one side or another, I'd avoid it, too.
Kilamanjaro Safaris should be fine, as you can brace yourself.
Similarly I had no real trouble with Dinosaur, even though everyone says it's really bumpy. I think it's because I didn't have trouble with up and down movements, as long as they were reasonably slow. That might be pushing it for you, though.
I didn't try the Rock and Roller Coaster, and I don't think I would.
Big Thunder Mountain and Everest were actually manageable for me, but only because the forces are more forward and back than side to side. However, you'll still be putting a fair bit of stress on your neck with these rides, so I wouldn't recommend them either.
Pirates of the Caribbean has a little drop, which wouldn't normally cause problems except that it's in the dark so you can't see it coming. Just be aware.
If you decide to chance riding something that might be hard on your back or neck (like Splash Mountain, which has a big drop at the end), remember to stay relaxed and look ahead in the direction you'll be moving. Drop your shoulders and let yourself bounce with the ride. Imagine yourself like a puppet with a string coming out of the top of your head, and just dangle from that. It really helps!
